Goldie's theorem
Encyclopedia
In mathematics
, Goldie's theorem is a basic structural result in ring theory
, proved by Alfred Goldie
during the 1950s. What is now termed a right Goldie ring is a ring
R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition
on right annihilator
s of subsets of R. Goldie's theorem states that the semiprime
right Goldie rings are precisely those that have a semisimple Artinian
right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem
.
In particular, Goldie's theorem applies to semiprime right Noetherian ring
s, since by definition right Noetherian rings have the ascending chain condition on all right ideals. This is sufficient to guarantee that a ring is right Goldie.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Goldie's theorem is a basic structural result in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, proved by Alfred Goldie
Alfred Goldie
Alfred William Goldie was an English mathematician.Goldie was Assistant Lecturer, Nottingham University 1946-48; Lecturer, then Senior Lecturer, Newcastle University 1948-63; and Professor of Pure Mathematics, Leeds University 1963-86;He won the 1970 Senior Berwick Prize from the London...
during the 1950s. What is now termed a right Goldie ring is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
on right annihilator
Annihilator (ring theory)
In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
s of subsets of R. Goldie's theorem states that the semiprime
Semiprime ring
In ring theory, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings....
right Goldie rings are precisely those that have a semisimple Artinian
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
.
In particular, Goldie's theorem applies to semiprime right Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
s, since by definition right Noetherian rings have the ascending chain condition on all right ideals. This is sufficient to guarantee that a ring is right Goldie.